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OPT ICS

1State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering,Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209,China. 2Centre for Micro-Photonics and CUDOS (Centre for Ultrahigh BandwidthDevices for Optical Systems), Faculty of Science, Engineering and Technology, SwinburneUniversity of Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia. 3Department ofElectrical & Computer Engineering, National University of Singapore, Engineering Drive3, Singapore 117576, Singapore.*These authors contributed equally to this work.†Corresponding author. E-mail: lxg@ioe.ac.cn

Pu et al. Sci. Adv. 2015;1:e1500396 2 October 2015

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10.1126/sciadv.1500396

Catenary optics for achromatic generationof perfect optical angular momentum

Mingbo Pu,1* Xiong Li,1* Xiaoliang Ma,1* Yanqin Wang,1 Zeyu Zhao,1 Changtao Wang,1

Chenggang Hu,1 Ping Gao,1 Cheng Huang,1 Haoran Ren,2 Xiangping Li,2 Fei Qin,3 Jing Yang,3

Min Gu,2 Minghui Hong,3 Xiangang Luo1†

Dow

The catenary is the curve that a free-hanging chain assumes under its own weight, and thought to be a “truemathematical and mechanical form” in architecture by Robert Hooke in the 1670s, with nevertheless no significantphenomena observed in optics. We show that the optical catenary can serve as a unique building block of meta-surfaces to produce continuous and linear phase shift covering [0, 2p], a mission that is extremely difficult if notimpossible for state-of-the-art technology. Via catenary arrays, planar optical devices are designed and experimen-tally characterized to generate various kinds of beams carrying orbital angular momentum (OAM). These devicescan operate in an ultra-broadband spectrum because the anisotropic modes associated with the spin-orbit inter-action are almost independent of the incident light frequency. By combining the optical and topological character-istics, our approach would allow the complete control of photons within a single nanometric layer.

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INTRODUCTION

As a quantum mechanical description of photons, optical angular mo-mentum is crucial for many classic and quantum applications (1). In1936, Beth first demonstrated that circularly polarized light (CPL) hasspin angular momentum of ±ћ per photon (2). Subsequently, in 1992,Allen and co-workers (3) recognized that helically phased beam withФ = lφ has quantized orbital angular momentum (OAM) of lћ, where lis known as topological charge. Since then, angular momentum freedomhas drawn increasing attention from many realms, including super-resolution imaging (4), optical micromanipulation (5), optical commu-nications (6, 7), and detection of rotating objects (8). Nowadays, lightbeam carrying OAM can be created by using various optical elements,such as spiral phase plates (9), computer-generated holograms (1),glued hollow axicons (10), form-birefringent elements (11, 12), mi-croscopic ring resonators (13), and two-dimensional (2D) nano-antenna arrays (the so-called metasurfaces) (14–19). In particular, themetasurfaces offer great advantages owing to their low profile and highdegree of integration. Owing to the spin-orbit interaction in space-variant anisotropic metasurfaces (12, 15, 19), CPL can be converted toits cross-polarization state along with the OAM generation, whereas thesign of the OAM is determined by the handedness of the CPL.When thespace-variant material is composed of half-wave plates with a rotatingoptical axis, the CPL can be completely converted (12, 15). However,the thickness of traditional wave plate is often much larger than theoperating wavelength as a result of the weak anisotropy (12).

Besides these inhomogeneous anisotropic media, it is also wellknown that chiral structures can interact strongly with CPL and mani-fest handedness-dependent phase modulation properties (20–25). The

handedness-dependent response of gradient metasurfaces can be con-sidered as one kind of scattering circular dichroism (26), owing tothe similarity between chirality and spin-orbit interaction. Never-theless, because of the resonant nature of chiral materials, it is difficultto achieve broadband performance.

In this work, we present a methodology that makes use of metallicstructures similar to the natural catenaries to generate a geometrical phasewith spatially continuous and spectrally achromatic distribution. The ge-ometric phase is solely determined by the surface structure because it re-sults from the spin-orbit conversion (12, 27) but not the phase accumulationalong the optical path (28) or impedance transitions (29, 30). On the basisof the unique features of a single catenary, we constructed arrays to gen-erate OAM beams with various kinds of phase distributions. The simul-taneous phase control over the azimuthal and radial directions suggeststhat our strategy is universal for almost arbitrary 2D phase profile.

RESULTS

As shown in Fig. 1A, we describe the optical catenary by the closed areaformed by two individual curves with a shift of D along the coordinateaxis. Such curves are inspired by the spin-orbit interaction in in-homogeneous anisotropic material (12) and rigorously derived by inte-grating the predefined phase function of space-variant shallow gratings(see Supplementary Text and fig. S1). Different from (12), the opticalcatenary does not require the anisotropic material to be a half-waveplate; thus, the thickness could be reduced to tens of nanometers. Owingto the constant phase gradient along the x direction, the correspondingcurve might be denoted as “catenary of equal phase gradient” similarwith the mechanical “catenary of equal strength,” which was first de-rived in 1826 by Davies Gilbert (31)

y ¼ Lpln jsecðpx=LÞjð Þ ð1Þ

where L is the horizontal length of a single catenary. In the catenary(aperture or its complementary structure), the inclination angle betweenthe curve tangent and the x axis, x(x), varies from –p/2 to p/2 betweenthe left and right endpoints, yielding a linear geometric phase as a result

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of the geometric relation Ф(x) = 2sx(x) (12), where s = ±1 denotes theleft- and right-handed circular polarizations (LCP and RCP).

The polarization-dependent geometric phases make the catenary intoa compact beam splitter for CPL, which is typically constructed by 3Dchiral structures (32). It should be noted that the geometric phase in cat-enary is free of discrete phase sampling as in the case of nano-antennaarrays (14). The elimination of discreteness implies that it is possible tomove an important step toward the metasurface-assisted law of reflectionand refraction (14, 33). Furthermore, the geometric phase in the catenaryaperture is intrinsically linear, whereas other deformed shapes, such ascrescents and parabolas, are characterized by large-phase nonlinearity,as can be directly calculated using the tangent angles (Fig. 1B).

In principle, owing to the gradient phase distribution, even a singlecatenary aperture can alter the propagation path of the CPL beam. Byarranging the catenaries in various forms, it is possible to obtain an ar-bitrary phase distribution, such as that featured by spiral phase plates,axicons, and lenses (Fig. 1, C to E). Without the loss of generality, threekinds of phased beams are considered: an ordinary OAM beam, a high-order Bessel beam (HOBB), and a focused beam carrying OAM. Theordinary OAM beam has a spiral phase along the azimuthal direction.The catenary curve in polar coordinates (r and φ) can be derived torepresent the corresponding surface topography (see section S1 of theSupplementary Materials for details)

r ¼ r0 þ mDð Þ jsec ðl − 2Þφ2

� �j� � 2

2−l ;m ¼ 0; 1; 2; 3… ð2Þ

Pu et al. Sci. Adv. 2015;1:e1500396 2 October 2015

where r0 defines the location of the vertex of the innermost curve, D isthe distance between two vertexes at adjacent curves, and m is the indexnumber of these curves. Following Eq. 2, one can use the catenaries toproduce OAM beams with arbitrary topological charge. As an example, asample with topological charge of l = −3 for LCP illumination (s = 1, thetopological charge is reversed from l to –l for opposite circular polariza-tion) was fabricated via focused ion beam (FIB) milling on a 120-nm-thickgold (Au) film (Fig. 2A), where values of D and r0 are 200 nm and 1.5 mm.

Under CPL illumination, the catenary apertures simultaneously gen-erate two kinds of beams with approximately equal intensity, one uni-formly phased and the other helically phased with opposite handedness(second column of Fig. 2A). On the basis of the interference effect, thetopological charge of the OAM can be directly identified, without the use

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r

BA

xy

(x)

Circular polarization

C

O r O r

D E

Catenary

x

0

ParabolaCrescent

rr

Fig. 1. Schematic representation of the geometric phase in opticalcatenaries. (A) Sketch map of a catenary aperture illuminated at normal

incidence by CPL. Light is assumed to propagate along the +z directionthroughout this paper. The inclination angle with respect to the x axis isdenoted as x(x). (B) Phase distributions of the catenary (red), parabola (or-ange), crescent (blue), and discrete antennas (black dot) for LCP illumina-tion (s = 1). (C) Three typical kinds of phase profiles for the generation ofpure OAM (left), HOBB (middle), and focused beam (right). (D and E) Ar-rangement diagrams of the catenary arrays for ordinary OAM (D) andHOBB or focused beam (E).

A532 nm

z = 27 m

m

l = –3 Ix Iy Itotal Ix Iy Itotal

632.8 nmz = 25 m

nmz = 20 m

306

l = –6

xy

C

532 nmz = 25 m

632.8 nmz = 22 m

632.8 nmz = 9 m

532 nmz = 8 m

nmz = 18 m

nmz = 10 m

l = 12

60

12

120

24

B

Fig. 2. OAM generators based on catenary arrays. (A to C) The topolog-ical charges for (A) to (C) are −3, −6, and 12 (s = 1), respectively. The first

column represents the scanning electron microscopy (SEM) images of thefabricated samples. Scale bar, 2 mm. The second column shows the spiralphase profiles. The last two columns depict the measured (the third col-umn) and calculated (the last column) results of Ix, Iy, and Itotal for l = 532 nm(top row), 632.8 nm (middle row), and 780 nm (bottom row). The distancesbetween the recording planes and the sample are indicated in each row.

Topological charge |l|

Co

nve

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n e

ffic

ien

cy (%

)

2 3 4 5 6 7 8 9 10 11 12 1310

20

30

40

50

60

λ = 632.8 nm

λ = 532 nm

λ = 780 nm

Fig. 3. Measured conversion efficiencies of the OAM generators.Mean values of repeated measurements are given for the samples with

l = −3, −6, and 12 (s = 1) at wavelengths of l = 532, 632.8, and 780 nm.The error bars give the maximum range of errors.

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of additional interference beam (1). In the experiment, we adopted threelaser sources at l = 532, 632.8, and 780 nm to investigate the broadbandperformances. The third column of Fig. 2A represents the intensitypatterns for different wavelengths and polarizations, which were recordedat a few micrometers away from the sample and in agreement with thetheoretical results obtained by vectorial diffraction theory (see the last col-umn of Fig. 2A) (34). For the x- and y-polarized components, the inten-sity patterns are manifested by rotating petals encircling the beam centers,where the modulus and sign of l are determined by the number andtwisting direction of these petals.

Pu et al. Sci. Adv. 2015;1:e1500396 2 October 2015

To validate the universal applicability of our approach, we fabri-cated and characterized two additional OAM generators with topolog-ical charges of l = −6 and 12 (s = 1). As we expected, the agreementwith theory is good except for some background noise (Fig. 2, B and C).Subsequently, we measured the conversion efficiency h, defined as theratio of the beam carrying OAM to the overall transmitted power(eq. S8). As depicted in Fig. 3, the mean values for l = 532, 632.8, and780 nm are 23.2, 39.8, and 54.4%, exhibiting at least 30-fold enhance-ment compared with that in circular nanoslits (16). In principle, theefficiencies can be further increased by using additional reflective layers

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z ( m)

x (

m)

0 10 20 30 40 50 60 70

10

0

–10

0

0.2

0.4

0.6

0.8

1

–10 –5 0 5 100

0.2

0.4

0.6

0.8

1

x ( m)

Inte

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0 10 20 30 40 50 60 70

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–10

z ( m)

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m)

–10 –5 0 5 100

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x ( m)

Inte

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ty

0

0.2

0.4

0.6

0.8

1

BA

D

E1.895 m

2.843 m

r1

r1

r2

r2

4 mC

Fig. 4. Catenary arrays for the generation of HOBB and focused OAM beam. (A and B) SEM images of the catenary arrays for HOBB (A) andfocused OAM (B). (C) Scaled view of the rectangular region shown in (B). (D and E) Measured cross-polarized intensity distributions of the HOBB (D)

and focused OAM (E) at a wavelength of l = 632.8 nm for RCP illumination (s = −1). The intensities at the xy plane (x, y ∈ [−20, 20] mm, z = 40 mm) areplotted in the insets.

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(17, 18). As an alternative to metallic catenaries, high-index dielectricmaterial can also be used to construct all-dielectric catenaries to enhancethe energy efficiency (15), although the operating bandwidth would belimited by the resonant nature.

Hereafter, we implement the catenary array to generate the HOBB andfocused OAM. The introduction of the Bessel-type radial phase distri-bution would enable diffraction-free transfer of OAM states, whereas thefocused OAM beam has potential application in subdiffraction imagingand micromanipulation. Following the design procedure in fig. S1 (B andC), another two catenary samples were fabricated (Fig. 4, A to C). Thepredefined phase distributions in the sample plane are as follows

FðrφÞ ¼ sðkrr þ l1φÞ ð3aÞand

Fðr;φÞ ¼ sðkffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ f 2

pþ l2φÞ ð3bÞ

where k = 2p/l is the wave number in vacuum. The design parametersare kr = k·arcsin(l/L), L = 2 mm, l1= 3, l2= 2, and f = 40 mm. The innerand outer radii, defined as r1 and r2, are 7.1 and 20.7 mm for theHOBB generator, and 10.6 and 20.8 mm for the OAM lens. The trans-mitted intensity patterns at l = 632.8 nm were measured using thehomebuilt microscopy under RCP illumination (s = −1). As shownin Fig. 4D, the intensity distribution along the propagation directionreveals the two main characteristics of the HOBB, that is, the diffraction-free property and the intensity singularity. The intensity map forthe planar OAM lens (Fig. 4E), however, is more constrictive along thepropagation direction. For the two kinds of devices, the measured center-to-center distances of the doughnut-shaped intensity patterns at z =40 mm are 2.843 and 1.895 mm, which are a bit larger than the theo-retically expected 2.54 and 1.56 mm.

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CONCLUSION

The most amazing property of the catenary-based OAM generators istheir ability to achieve achromatic performance, which can be con-cluded from the following two observations. On the one hand, the ge-ometric phase is intrinsically independent of the operating frequencyaccording to its definition (F = 2sz). On the other hand, the conver-sion efficiency h is also nearly achromatic, especially when the char-acteristic dimension is at deep-subwavelength scale. To demonstratethe latter conclusion, the efficiencies for both the catenary aperturesand discrete antennas were calculated using commercial softwareCSTMicrowave Studio, where the unit cells (cells A and B) were chosenas shown in Fig. 5 (A and B). To increase the accuracy, the measuredpermittivities for gold and SiO2 substrates were adopted (35) in thenumerical simulations. It should be noted that the width w is changingall over the catenary; thus, the period of cell A (p = 2w) varies corre-spondingly (Fig. 5A). As can be seen in Fig. 5C, the efficiencies of cellA for p = 100 to 400 nm are depicted in a region shown in gray (thelower envelope of this area corresponds to the lowest efficiency),whereas the results for cell B are evaluated for only p = 400 nm. Ob-viously, the operating bandwidth of the catenary is far beyond that ofthe discrete antennas, especially in the low-frequency regime. It should benoted that the simulation results obtained from the unit cells representonly an approximation to the real space-variant transmission. Neverthe-

Pu et al. Sci. Adv. 2015;1:e1500396 2 October 2015

less, this method provided a simple way to predict the electromagneticresponses, which was widely used in related research (15, 17, 18).

To explain the physical mechanism of the achromatic h, the elec-tromagnetic modes for the two unit cells are shown in Fig. 5 (D andE). Resorting to the optical nanocircuit theory proposed by Engheta(36), we found two orthogonal modes in the catenaries, one inductiveand one capacitive, along the main axes. Because the inductive andcapacitive modes are related to the achromatic rejection and transmis-sion, one can expect that h is nearly a constant in an ultra-broadbandspectrum (see eq. S8). The fluctuation in the visible region is becausethe unit cell of the catenary has a period close to the operating wave-length; thus, there is an obvious amplitude modulation effect. Bytreating the catenary as a space-variant grating, we evaluated the trans-mission coefficients at different locations for both the copolarized andcross-polarized CPL. As shown in fig. S2, both transmission coeffi-cients are dependent on the width of the catenary, and the fluctuationof the copolarized light is a bit larger than its cross-polarized counter-part. At the low-frequency regime, the amplitude modulation effect ismuch weaker, because the period is in the deep-subwavelength scaleand the metals behave more like perfect electric conductors (14). Asa result, we can expect that if smaller structures are adopted, the am-plitude variation can be eliminated.

The periodic change of the amplitudes has two effects: first, forcross-polarized light, the generated field distributions will depart fromthe ideal case, owing to the additional amplitude modulation; second,for copolarized light, the periodic variation of amplitudes will lead toobvious diffraction. To illustrate these phenomena, we measured theintensity patterns of both the cross-polarized and copolarized compo-

A B

C

p

s

p/2 p

p w = p/2 D

u-direction:inductance

v-direction:capacitance +inductance

Cell A

u-direction:capacitance +inductance

E

Zu = j L1 + 1/j C1Zv = j L2 + 1/j C2

Zv = j L + 1/j C

Zu = j L0

u

v

Co

nve

rsio

n e

ffec

ien

cy

(%)

Frequency (THz)0 100 200 300 400 500 600

0

10

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30

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50

60

70

80

90

100

p = [100, 400], Cell A p = 400, s = 350, Cell B p = 400, s = 300, Cell B

Cell BUnit cells

Measurements

EinEv

Eu

Fig. 5. Comparison of the conversion efficiencies and electromag-netic modes of the catenary apertures and discrete nano-antennas.

(A and B) Schematics of the unit cells in catenary apertures (A, s = p) anddiscrete antennas (B, s < p). The two cells differ from each other in thelength of the metallic bar with respect to the period. (C) Numericallycalculated polarization conversion efficiency h for the catenary aperturesand nano-antennas. The small discrepancy between the theoretical expec-tations and the experimental results (Fig. 3) is mainly due to fabricationimperfections. (D and E) Schematic of the orthogonal modes in the u-vcoordinates for the catenary (D) and discrete antennas (E). The surface im-pedances as a function of frequency are depicted by the nanocircuitmodel. The resistances stemming from the ohmic loss are neglected.

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nents for a sample designed to generate OAM of l = ±1. As shown infig. S3, the cross-polarized intensity departs slightly from the calculateddoughnut shape, whereas the copolarized component results in a trian-gular diffraction pattern (details are discussed in the SupplementaryMaterials).

In summary, we have theoretically predicted and experimentally dem-onstrated the intriguing properties of the optical catenaries, which arestructures well known in the natural world and widely exploited inarchitecture but still unfamiliar to optical researchers (37, 38). The perfectOAM demonstrated here, with spatially continuous and spectrally achro-matic phase distribution, will allow a revolution of the state-of-the-arttechniques with applications such as optical micromanipulation andhigh-speed photonic communications. As a direct extension ofthese findings, novel optical functionalities such as giant enhancementof polarization selectivity (32) and achromatic light routing (39) couldbe enabled.

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MATERIALS AND METHODS

Numerical simulationThe unit cells were simulated using the finite element method in CSTMicrowave Studio. Two Floquet ports are placed at the +z and −zdirections, respectively. The port at the −z direction is attached tothe bottom of the SiO2 substrate, which has a thickness of 1 mm.The distance between the sample surface and the port at the +z direc-tion is also set as 1 mm.

The propagation of the OAM beam was simulated by using vec-torial angular spectrum theory (34). In all the calculations, we assumethat light is propagating along the +z direction. When the electric fielddistribution at z = 0 is known, the vectorial electric field at any planewith z > 0 can be written as follows

Exðx; y; zÞEyðx; y; zÞEzðx; y; zÞ

24

35 ¼ ∫

∞

−∞∫∞

−∞

Ax;zðm1; n1ÞAy;zðm1; n1Þ

m1Ax;zðm1; n1Þ þ n1Ay;zðm1; n1Þ−q

2664

3775

� exp i2pðm1xþ n1yÞð Þdm1dn1 ð4Þand

Ax;zðm1; n1ÞAy;zðm1; n1Þ� �

¼ exp i2pqðm1; n1Þzð Þ� Ax;z¼0ðm1; n1ÞAy;z¼0ðm1; n1Þ

��

¼ expði2pqðm1; n1ÞzÞ

�∫∞

−∞∫∞

−∞

Exðx; y; 0ÞEyðx; y; 0Þ

��expð−i2pðm1xþ n1yÞÞdxdy

�ð5Þ

where m1 ¼ kx=2p; n1 ¼ ky=2p; q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=l2 − m2

1 − n21

qare the spa-

tial frequencies, kx and ky are the wave numbers along the x and ydirections, l = l0/n is the wavelength in free space, and n is the refractiveindex of the background medium.

Sample fabricationAll the samples were fabricated on 1-mm-thick quartz substrates. A3-nm-thick Cr film and a 120-nm-thick Au film were subsequently

Pu et al. Sci. Adv. 2015;1:e1500396 2 October 2015

deposited on the cleaned substrates by magnetron sputtering in a samesputter chamber. The sputter deposition rates for Cr and Au were RCr ≈0.83 nm s−1 and RAu ≈ 0.92 nm s−1, respectively. The Cr film was usedto improve the adhesion between the Au film and the substrate. Catenaryarrays were then milled on the Au/Cr film using a Ga+ FIB (FEI HeliosNanoLab 650), and the accelerating voltage and current of the Ga+ beamwere set as 30 kV and 24 pA, respectively.

Optical setupAll measurements were performed on a homebuilt microscope in trans-mission mode. First, the incident collimated beam was converted into CPLvia the combination of a polarizer and a quarter-wave plate. The trans-mitted intensity patterns were imaged by using a 60× objective lens anda tube lens, and collected by a silicon-based charge-coupled device (CCD)camera (1600 × 1200 pixels, WinCamD-UCD15, DataRay Inc.). For themeasurement of the OAM generators, the interference patterns werefiltered by a rotatable polarizer before the CCD camera to obtain the x andy components. For the measurement of copolarized and cross-polarizedtransmission, a quarter-wave plate and a polarizer were used.

SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/1/9/e1500396/DC1TextFig. S1. Schematic of the integration procedures to design the catenaries.Fig. S2. Transmission coefficients of the subwavelength grating.Fig. S3. Design, fabrication, and characterization of the sample with l = ±1.

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Acknowledgments: We thank L. L. Yang, H. Zhu, X. C. Lu, and W. B. Xu for their helpful dis-cussions. Funding: X. Luo acknowledges the financial support by the 973 Program of Chinaunder contract no. 2013CBA01700 and the National Natural Science Funds under contact no.61138002. M.G. thanks the Australian Research Council for its support (FL100100099 andCE110001018). M.H. acknowledges the CRP project: “National Research Foundation, PrimeMinister’s Office, Singapore” under its Competitive Research Program (CRP award no. NRF-CRP10-2012-04). Author contributions: M.P., Xiong Li, and X.M. contributed equally in thedesign, fabrication, and characterization of the sample. M.P. and X. Luo wrote the first draftof the manuscript. M.G. and M.H. provided major revisions. All authors discussed the resultsand commented on the manuscript. X. Luo proposed the original idea and supervised theproject. Competing interests: The authors declare that they have no competing interests.Data and materials availability: All data, analysis details, and material recipes presentedin this work are available upon request to X. Luo.

Submitted 27 March 2015Accepted 17 August 2015Published 2 October 201510.1126/sciadv.1500396

Citation: M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li,F. Qin, J. Yang, M. Gu, M. Hong, X. Luo, Catenary optics for achromatic generation of perfectoptical angular momentum. Sci. Adv. 1, e1500396 (2015).

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Catenary optics for achromatic generation of perfect optical angular momentum

Haoran Ren, Xiangping Li, Fei Qin, Jing Yang, Min Gu, Minghui Hong and Xiangang LuoMingbo Pu, Xiong Li, Xiaoliang Ma, Yanqin Wang, Zeyu Zhao, Changtao Wang, Chenggang Hu, Ping Gao, Cheng Huang,

DOI: 10.1126/sciadv.1500396 (9), e1500396.1Sci Adv 

ARTICLE TOOLS http://advances.sciencemag.org/content/1/9/e1500396

MATERIALSSUPPLEMENTARY http://advances.sciencemag.org/content/suppl/2015/09/29/1.9.e1500396.DC1

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